 So thanks a lot for the invitation to this conference as you can see I'm a mathematician I'm working in a math Institute, and I would like to present some recent ways of deriving effective equations for many particle system. I put this word derivation here in a quotation marks for one reason. I have to say I don't like this word with this in this context The point is when mathematicians talk about Deriving an equation they not really mean to derive the equation they mean to prove the validity of this equation with mathematical rigor So they do the mathematical rigorous part of an equation which has been derived by physicists many years ago And I use this word here because that's the way Mathematicians like to talk about it So if you read an article or some colleague tells you I have derived some equation and he's a mathematician To avoid misunderstandings. Well, I also use it because that's the common language. Although another word would be better. I Would like to start with a motivation and I'm interested in systems of Many particles and in particular of the dynamics of such many particle system in particular when there's Interaction between the particles and for many cases, you know, you're the experts. It's difficult to have some analytical or numerical Result for that. So if you really for example start with the many body Schrodinger equation and really Want to simulate the Schrodinger equation not some approximation the true Schrodinger equation with an interaction term This is numerically often impossible when you have a few particles So in the quantum mechanical situation of the particle numbers, let's say 10 or larger and you have All kinds of correlations between the particles. This is usually hopeless and what you do as this is this I heard that already in a talk today. You use some kind of simplified Effective description you replace the true interaction by some kind of effective for example mean field Equation and this is what you do now It's a very active field now in mathematical physics is to prove the validity of such effective description So you start with these Effective equations and you prove rigorously that they are really close to the true situation where you look at the true microscopic Equations of motion and what is particularly interesting of course is the interaction when there's no interaction It's usually easy to do when you have independent motion of the end particles There's not a lot to discuss, but when you have interaction between the particles This is interesting as I told before I'm interested in the dynamical questions also in this talk in particular for quantum Situations you could also ask static a question you could ask the question How does the ground state look like for system of n particles with interaction and Ask about the ground state. So this is a static question This is also not easy to do that one also uses effective descriptions But here I'm interested in the dynamics I want to mention a few examples where such things might be interesting. These are all examples We do research on in my group with my group in Munich For example, you could say you want to derive the Hartree-Fock equation To describe the dynamics of large molecules when chemists describe the dynamics of large molecules with the computer numerically They always use Hartree or Hartree-Fock, but the true description is of course the n-body Schrodinger equation with paint action or the more fundamental Description and of course it would be interesting to start from there In Kolb Wozegase, there's many results in the mathematical literature You can derive for Hartree or physically even more important the so-called cross Peter Yasky equation You might have heard about this cross Peter Yasky equation. It's not really a mean field equation I would say it's the point is to derive if you make a mean field ansatz You arrive at some cross Peter Yasky equation with the wrong coupling constant Compared to what you see in the experiment and in fact, there's pair correlations between the particles So the particles usually it describes the physics of a dilute bozegas at Condensation and the particles since the gases dilute don't hit so often but when they hit the Collisions are comparably hard and you have some pair correlations You have to consider the correlation with the next nearest neighbor So to say to get the right coupling constant out So it's not really a mean field a situation although many people claim it would be and there's if you do it rigorously Of course, you have to really consider this pair correlations and this is something that's that's many results well just one Note the for also the effective equation for fermions are so important There's very few mathematical results. So comparing to bosons boson. There's a lot understood fermions are mathematically here Very hard to handle What we also did we derived Maxwell's equation from some QED model as You know, it's some kind of semi-classical limit. So to say, you know, you should describe also Electromagnetic effects and with photons and This is not so easy to do of course if you have classical behavior It's clear that the Maxwell's equation come out because these are the Heisenberg's equation But to really show that there's classical behavior. It's not easy to do and what you can do is if you go to a many body system It is known that if the photons are in a coherent state they React classically on the system But what is the mathematically challenging part is to show that the photons which are naturally Created by a system are close to a coherent state They will never be exact in a Korean state a coherent state is an idealization So what you want to do is you start with a gas of charged particles You show that as time evolves the photons created are in some sense close to a Korean state And then you show that this closeness is sufficient to have a classical behavior as a Back reaction on the charges and this is something which you can do So you can also read the Maxwell's equation as some effective description for QED in a certain situation Of course where they hold today, I would like to focus on Something one could call the most obvious system or something which has been discussed for a long time Most obvious in the sense this might be the first physical system you heard about in school It's just a Newtonian particles with interaction. This is a conference on long range interaction and Therefore I would like to exclude collisions. So collisions are very hard to treat with mathematical rigor So we really think of Particles which interact on a longer range So I would like to Introduce now the notation I will use before I say a little bit more about the technicalities and what people prove here So you have an interacting particles as I said for example stars and Newtonian dynamics So these equation which I would like to derive now derive is Important for the dynamics of Galaxies or plasmas so when I talk about particles you could have the electrons in some plasma in mind For example in some fusion reactor you want to describe the plasma You don't describe the motion of all the electrons, but you describe the cloud somehow or what you can also have in mind You can think about stellar dynamics You have a galaxy formed as stars you go to a regime where Relativistic effects are not so relevant and you describe the motion of the cloud of stars here I learned recently that it's even more interesting if the points are galaxies and you go to Structures which are even even beyond the galaxy structure But I will talk in the in this talk. I will always refer to stars So our points are some stars in some kind of a galaxy they move we assume that it's a good approximation to do Newtonian dynamics and by gravity you have pair interaction but remember the system might be interesting in other situations and Describe it by some trajectory in phase space. So this is the notation I will use q for the position p for the momentum I will Set the masses to be equal to one so p and it's the same as the velocities or q's Position p velocity and I have n particles and qj is the position of particle number j pj Is the momentum or the speed of particle j? You totally dynamics this you know the time derivative of q is given by the velocity And the time derivative of the velocity if given given by the force acting on the system We could introduce an external force This would not cause any technical problems, but at the moment I just assume that I have an interaction force so the force between the two particles is given by This pair interaction here So here I have the force between qj and qk and I want to look at the situation where the forces of order one in my scaling so I To have the following picture I have a box of fixed volume and I put more and more stars into this box and I look about at the interaction which Is such that the interaction strength keeps constant and this I can make sure by this Prefactor into the minus one which counts as the summation you might say this looks unphysical well It looks but it is not because you can say well It's just a rescaling of the true situation if you say okay I will have a volume which increases with the particle number proportional to the particle number and you rescale space and Time in the respective way you can get rid of this pre factor and have a system Which makes more sense because of in physics it doesn't make so much sense to change the interaction behavior the coupling With the particle number, but as I said, this is just a rescaling so it can be physically argued for And now I want to derive a macroscopic law of motion Yeah for what is always a question, which is the macroscopic optic you want to describe Well, I would like to describe the particle density and how I do this there I give now a picture which might be very easy, but I think it's good to have pictures to keep in mind And this will be relevant for later because I will now form the galaxy though this I will use later I will let's think we have some empty space We play God so to say we take the stars and throw them into this empty space forming a galaxy And what I will assume is that I throw in these stars in an IID way So in an I have an initial Probability density row zero some nice and smooth probability density on the one particle Face space and I throw in the particles in an IID way So my first particle from my first star will sit on a certain position q1 p1 The second somewhere else and then as time goes on I form this galaxy with stars sitting in certain position Now what is the density? Well the density to discuss about the density you can say let's take a volume V Which is an on an intermediate scale So the volume is small compared to the volume of my full galaxy But still large enough that it contains many stars So the number of stars will be very large and we will contain many stars But still be small and then you can move this V around and then you get this density and Now I would like to show what is the time evolution of the system So now if I look at this I can say well I want to guess the force Which is felt on the position q so let's assume there's a star I painted it in red here sitting on the position q and I want to know what is the force this guy Experiences the first of all of course I have this coupling constant n to the minus one Which was my coupling constant and then I have the sum of all these boxes So I take J is the label for the different boxes and I take the force coming from all the Particles or all the stars in one box times the number of stars which there are in this box and I get the simple formula Well using my definition of this density of course the density is defined by the number of particles in the box Divided by V times the total number of particles. This is the definition of the density and if I plug that in This is not If I use that you get this formula and this formula Can be approximated by an integral so I have this volume element vj And you can write from the sum times the volume into a integral It turns to an integral over this volume here this remember this volume is a volume in face space It's a six-dimensional volume and I arrive at this formula This here is in good approximation You can read this Well, this is a good approximation because it's the respective theme on some and this year I use the notation. This is a convolution the q variable if you look at this you integrate out p and then you make a Covolution in q so you have q minus qj not p minus pj You only have the q minus qj so you make a convolution in q and an integral in p and this is the notation I use So we have guessed now this kind of formula for the system and here It's very important to note that one argument which I used here is the law of large numbers argument Why is it included here? Well, I had a probability density row Which was given initially and I threw the particles in an iid way into my cloud Another law of large numbers argument tells us that the empirical density so the density Which is empirically there which describes where my particles are sitting is close to this probability density I think this is very important because Conceptually these are very very different things and empirical density and the probability density Although the formula is the same because of law of large numbers So here the row turns into empirical densities Therefore there's really a force acting on the star which is there and it's not only some expectation value of a force which could Be larger or smaller and just in mean hit the right thing. No, it's really the force which there is Which we which we have estimated here Now of course the system is dynamical with the motion of the particles also my row changes but as you know So this row will be time dependent because of the motion of my my stars here But you have conservation of face-based volume this you all know so if you move with one of these boxes The number of particles which remain in our box will be a constant and therefore what you get is that if you Estimate your row along such a trajectory So follow the trajectory of the particle sitting in the center of the box This will be constant because little n is now a constant and large n and V are constants anyway Which means that the time derivative of this guy is zero and you arrive at the respective continuity equations of the partial derivative plus the spatial derivative times q dot plus the Velocity derivative times p dot is equal to zero and now we know what we did before we said well this p dot comes from this Force and I have a good approximation now for this force and this I can plug in so here I have this if bar and if you remember what I got for this if bar I get this equation So what we have here is now the so-called blast of equation guessing it Euristically so therefore why there was no real rigor mathematics. It was just like a physicist would do it, right? This is the way you would have done it and Later I would like to talk about a rigorous proof Can you really show with mathematical rigor that this describes the situation before I do that? I would like to mention the following. This is a nonlinear equation the row It's in a nonlinear way here row times something which depends on row And this comes from the fact that I have this many body system and that this Density turned into an empirical density. There's also nonlinear Schrödinger equations where you have some nonlinear term and Sometimes people use them if you only have a few particles So you have assume you have a nonlinear Schrödinger equation For the interaction term is nonlinear. So you have an interaction row times In convolution with some V This is a very strange thing because you have a probability density in convolution with some potential. So you have It seems like you think there's an interaction Although there's only the possibility that there might be a particle and if you only a few particles The such equations do not make sense Let's say they cannot be explained for or derived from the linear Microscopic system, but here it's a little bit different because here you should understand I am in the many body system So my probability density is not only a probability density. It's now an empirical density It's an density of particles which are really sitting there and now to say from them I have this interaction makes perfect sense Okay, I have this philosophy equation and What I would like to do next is I would like to talk about some mathematical results and This was a big question as I said It's a very active field in mathematical physics to derive effective equations and this was a big question was Discussed already in the 70s. There's rigorous results with this It's really proven that the true density if you follow the Newtonian dynamics with panta action is close to the Solution of the glass of equation. I would like to say a few words about the sense of closeness in a second Going back from to the 70s. Yeah, I mentioned two important results and There's has been many similar results until today and the problem now is As you can see that they assume that f is globally lipchits. So the interaction force is Globally lipchits in particularly bounded, but as I said in the beginning the interesting physical cases are plasmas or Galaxies and there you have Coulomb interaction repulsive or attractive so gravitation or with the pulse of Coulomb and that's of course not included in these papers What they do is Typically, well, there's also some other approaches, but in the most well Way which is done very often is that you understand this trajectory X as a density Remember what I said. So I have the one the microscopic system. The microscopic system is described by a trajectory X large X Consisting of Q and P the trajectory and phase space the effect is description is now evolution equation for density These are completely different objects that trajectory and a density and before I be able to compare them I have to bring them on the same level and one way of bringing them to the same level is to say okay I translate my trajectory into a density by looking at this object I take the sum over all these data functions So then I have the data functions on the positions where the particle are then it's a density And what the people do is they show Some deterministic result. I will say something about the word deterministic soon They will show that if initially at time zero my empirical density So the sum of data is close to my row zero Then this holds also at time team Whereas this here I have the Newtonian dynamics of this trajectory X and build the empirical density at this later time And here I have row zero to row T. This is the solution of this loss of equation, which I have guessed your stick Deterministic means here now what they do is they can show when if ever this is closed then this is close no matter what and I will give a probabilistic result later I will show that this is not true always, but with very high probability The details will come next therefore. I call this a deterministic result Of course the closeness for those who have some mathematical training It's not in L one of course the one function is a sum of deltas the other is smooth No, it's a closest in sense of yeah some observables. You have some nice and smooth Function on face base you use this is an observable and then you compare the smooth density to the sum of deltas And then of course it makes sense to have some closeness some matching and this is what one does here mathematicians call that Wasserstein distance, but this is something which is for you very easy to understand But I'm not writing it here because this is not what I will do later in the talk Now as I said the interesting cases are who long and Then you're of course the forces q over q3 So I dropped the coupling constant You know we mathematicians set all the factors always equal to one to have less have some ease of writing So the coupling will just be some constant and there was now a reason result And if you look at that so it started in 74 and the next really I would say Interesting step forward to get closer to Coulomb was 40 years later And this is a result by uray and jabon and what they consider they do not consider the Coulomb case So they take a singularity which is a little bit weaker. They have this minus delta here data They can show for any positive delta. They can handle these potential So it's slightly weaker than Coulomb and they need a cutoff at n to the minus 6 1 6 why I have the Explanation what a cutoff is here what you all know that so I don't have to say anything here, and this is what they do And I would like to mention here at this point that doing the rigorous derivation of the Vlasov equation for the full Coulomb singularity without any cutoff This is still an open problem until today and I think it Would be very nice to get closer to that and this is now what I will do later in this talk to show You how one can improve these things what they do and this is now I said the other stuff was Deterministic this is now probabilistic in some sense now what arrange a burn do they exclude some particular Initial conditions, so there's some initial conditions where they say they don't get This this this result here But they can show that these initial conditions are Untypical untypical means they have small probability Remember the picture I had at the beginning I form my galaxy throwing in stars in an IID way So I get a probability measure and with respect to this probability measure the initial condition they exclude have small probability So it's still a very nice result Now you could say well it could be There's always if you do rigorous stuff the question is this condition really Needed or is it just technical a technical condition is something where it's just in fact You don't need the condition, but without the condition It's just too hard to prove something the statement is still true But you don't know how to prove it This is a technical thing and it could be that this is technical But I can show you know an argument that it's not technical that in general this is wrong Which also means that this additional Assumption they have that they exclude some some things that this is not bad So it looks like a beaker result But in fact it's it's not the case because the stronger result if you would just say okay I assume that this holds for this interaction. It's wrong and I will give you the argument and the argument is as follows Consider the repulsive case so the argument works best in the repulsive case You have a repulsive system and you take your n particles which you have let's say an electrons and you form clusters in Each clusters there's epsilon times n particles and the number of clusters is epsilon to the minus one and will be Extremely large and epsilon will small such that epsilon n will still be very very very large This is the idea which you have and now you form this clusters Since you still have many clusters namely epsilon to the minus n if you look at it From far away with the blurred lens so to say you still see a nice and smooth density So it still converges in some weak sense in some if you look at observables against the smooth row zero So you still have initial convergence, but now if you look at the time evolution of the system what happens well I have many many particles in the cluster Almost of order n so this n to the minus one Prefactor from the coupling is cancelled the only the epsilon is left But I have the singular behavior So if the singularity is such that the cluster is so that I'm so close the particles so close to each other that the Singularity is stronger than this this factor epsilon Then the potential energy for the particles will be very large So what I will have here I can find by tuning the value for epsilon Find a situation where the cluster such that the potential energy of all these particles is very large And what will happen now is very clear. So these clusters they will explode The particles in particular those on the edge They will be propelled away from the center and after some time the system will look like this You will have particles which are very fast move in all kinds of direction They will go out very fast and will not be described by this nice laws of equation which gently Evolves in phase space. So it's clear that for such a situation Things turn out bad, but now remember I said it's small in probability to form These clusters of course in an iid way if you throw in your particles iid to have these very close clusters is very Improbable, so it's in line with what I said before Now what I want to tell you next now in the next in the remaining five minute remaining five minutes Is about another approach which I have found with my students and what we do is we start in a different way We again you remember I have to control a trajectory with a density and What usually people do is they Translate x into a density I do the other way round The idea is to bring road to the level of trajectories and I would like to explain you how one can do that I Assume that the blast of equation has some solution so I started again I have this row zero I form my galaxy and the first thing I do is before looking at the trajectories I have solved the blast of equation and this solution now is given Given this solution. I now define an Acciliary system in the following way again. It's a Newtonian system q dot is given by p p dot by some force But this time I don't use the pair interaction force No, I assume for each particle that each particle feels this force which I have guessed initially and now the Good thing is here. I start of course with the same initial condition as in my true system and What the goal is now to prove that my true in my auxiliary system match the q and q bar and p and p bar are always close There's one very important Thing which I would like to note Here I don't have pair interaction. I have this is like an external field So the independence I had initially is not lost this time evolves. This is the big advantage And now the goal is to compare q and p So this I would like to mention the results which we have this was one of my students and We look at we first started also with a force which is slightly weaker than Coulomb But the cutoff is much better. It's the n to the minus one third compared to n to the minus one six This is interesting because n to the minus one third is the distance to the next neighbor in average, right? You have a volume of order one the next neighbor has Typically the distance n to the minus one third we could improve this result and shift so to say the data to there So we could look at the true Coulomb force with the price. We had to pay us. We had to make the cutoff a little bit worse But but it's interesting because it's the Coulomb I would like to talk only about the first thing because it's a little bit easier to explain Now remember this qj's the initial positions and momentum of our IID Though I get some probability measure on the phase space my random object is this initial initial Situation so this large x0 is that what you usually know as little omega and I have a product measure on this space Now these queues they are functions of the initial Situations are given x0 queues the queues in peace can be can be determined So they are random variables and now the interesting thing is for my auxiliary system Which I have introduced is Independence is not lost and the probability density is just transported by the laser flow after some time t the Probability density if given by the solution of lassoff And now what the law of large numbers argument tells us now since after t I still have law of large numbers that my q bar P bar system they will that will always converge against row This is just law of large numbers the direct application So all I have to show is that my true system is close to this auxiliary system Then I know that also my true system describes the solution of lassoff Now comes the part which is a little bit more technical now This is like an unsat I will introduce now a few objects which might look strange and on the next slide I will explain you the advantage of these objects and why I use them So this is now the set I'm looking at this is the set of bad trajectories Trajectory is bad when the deviation of the true and this auxiliary Solution is larger than n to the minus one third you might still say well n to the minus one third This is not very much. Well, I'm saying my I have high standards here at the moment And I will tell you why these high standards are in fact helpful So we have very high standards We already call something bad if only one particle here I have the maximum norm so if only one of the particles deviates more than n to the minus one third and I say well This trajectory is bad already and what we prove as a theorem is we show that for any team if I take the limit n to infinity the particle number n to infinity that the probability to have a bad trajectory is Equal to zero in this limit and I we can also do well very often when Mathematicians make these derivations they have results like that, but I think it's very Important to get error estimates to really be able to talk to physicists And I can give the error estimate which is also striking the error estimates So this probability is bounded by c gamma n minus gamma for any gamma for so for any gamma You give me I can find a c gamma that this probability is bounded by that so it goes down faster than any polynomial What do I do how do I prove that and I would like to give you just a few arguments how this is done? I define this Other random variable. It's again this distance between true and auxiliary trajectories Multiplied by n to the one third, but I cut off this value if it's one So if this gets larger than one, I just keep the Value one which means that this J is One if and only if my my my w my is in this 80 So if if I'm in the set 80 only if I'm in the set 80 then this value is equal to one And then what I will prove and I will give you some ideas how to prove it on the finest light I will prove that the time derivative of the expectation value of J is bounded by the expectation value of J Plus something which is small as n tends to infinity if I have this Most of you might know green ball lemma then by green ball you find this Equation and of course at this expectation value at time zero is equal to zero because initially both things matched and Since J is positive you can show that the probability to be in a is bounded by the expectation This comes from this fact you get the value one whenever you and a when you're not in a you get something positive So the expectation value is a larger than the probability to be in a and therefore if I can control e Then I can show that my theorem holds now. This is the last slide I would I recall here this end and I would like to Explain you why we use this J and what are the advantages? This will not be technically I just Euristically explain the advantages and the point is you could say well why not just do the following Take this probability to be in a p of a and try to find a green ball inequality for that So look at the time derivative of p of a and so on well the argument is the following an expectation value is easier to control why The expectation value of the characteristic function of a is the same as the probability of a that you know Remember our a was this set this this set here and The point is if you look at this now This J isn't like a characteristic function whenever I'm in the set a I get the value one But outside a I'm not putting it to zero. No, I smoothly Go up to one so I have something like a smoothed out Probability to be in a this of course has advantages if you look at the time derivative and some if you have a characteristic function So the probability and the trajectory goes down from one to zero all of a sudden This is of course bad to control But if the value goes down gently and this is what my J does it's much easier to handle and from the technical review you might Believe me that this is very very helpful So this J has reached its maximum whenever this is equal to n to the minus one third It cannot grow anymore and remember I wanted to prove some green ball estimate I wanted to estimate the time derivative of the expectation value of J Well, then I have reached the maximum. I cannot grow any further I can only go down and I wanted to have a bound from above which means when my trajectory is bad already I don't care anymore. What happens in the future? It just assume well it stays bad and to take the value one But the time derivative of the expectation value which measures the badness can only be negative It can only get better Therefore, I do not need to consider these cases the estimates for those cases are trivial There the time derivative is negative. I get a trivial upper bound and This is of course now related to what I said at the beginning somehow I have to exclude this clustering And now remember this picture with the clusters. How do I exclude it now? I look at my auxiliary system the Q bar P bar system there I can do law of large numbers argument So these clusters there their probability is so small that you can forget about them So for the auxiliary system, there's no clustering now my true system I only have to consider cases where my Q bar P bar and Q P Close to each other all the other cases are not interesting and now I give you the following Exercise take an Q bar P bar, which is nice and smooth Yeah Particle is no clustering and now you may shift each particle by end to the minus one third and try to form a cluster Of course, that doesn't work. So no matter what you do You can really exclude clustering rigorously using the law of large numbers argument on Q bar P bar only where you can use it and argue there's no clustering and then do the technical stuff and you have excluded all the bad Systems so I wanted to give an outlook, but I think I skip that because time is over and I would like to thank you for the attention and Yeah